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Dave Reed

Dave Reed. Connectionist approach to AI neural networks, neuron model perceptrons threshold logic, perceptron training, convergence theorem single layer vs. multi-layer backpropagation stepwise vs. continuous activation function associative memory Hopfield networks, parallel relaxation.

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Dave Reed

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  1. Dave Reed • Connectionist approach to AI • neural networks, neuron model • perceptrons threshold logic, perceptron training, convergence theorem single layer vs. multi-layer • backpropagation stepwise vs. continuous activation function • associative memory Hopfield networks, parallel relaxation

  2. Symbolic vs. sub-symbolic AI • recall: Good Old-Fashioned AI is inherently symbolic Physical Symbol System Hypothesis: A necessary and sufficient condition for intelligence is the representation and manipulation of symbols. • alternatives to symbolic AI • connectionist models – based on a brain metaphor model individual neurons and their connections properties: parallel, distributed, sub-symbolic examples: neural nets, associative memories • emergent models – based on an evolution metaphor potential solutions compete and evolve properties: massively parallel, complex behavior evolves out of simple behavior examples: genetic algorithms, cellular automata, artificial life

  3. Connectionist models (neural nets) • humans lack the speed & memory of computers • yet humans are capable of complex reasoning/action  maybe our brain architecture is well-suited for certain tasks • general brain architecture: • many (relatively) slow neurons, interconnected • dendrites serve as input devices (receive electrical impulses from other neurons) • cell body "sums" inputs from the dendrites (possibly inhibiting or exciting) • if sum exceeds some threshold, the neuron fires an output impulse along axon

  4. Brain metaphor • connectionist models are based on the brain metaphor • large number of simple, neuron-like processing elements • large number of weighted connections between neurons note: the weights encode information, not symbols! • parallel, distributed control • emphasis on learning • brief history of neural nets 1940's theoretical birth of neural networks McCulloch & Pitts (1943), Hebb (1949) 1950's & 1960's optimistic development using computer models Minsky (50's), Rosenblatt (60's) 1970's DEAD Minsky & Papert showed serious limitations 1980's & 1990's REBIRTH – new models, new techniques Backpropagation, Hopfield nets

  5. Artificial neurons • McCulloch & Pitts (1943) described an artificial neuron • inputs are either excitatory (+1) or inhibitory (-1) • each input has a weight associated with it • the activation function multiplies each input value by its weight • if the sum of the weighted inputs >= , • then the neuron fires (returns 1), else doesn't fire (returns –1) if wixi >= , output = 1 if wixi < , output -1

  6. Computation via activation function • can view an artificial neuron as a computational element • accepts or classifies an input if the output fires • INPUT: x1 = 1, x2 = 1 • .75*1 + .75*1 = 1.5 >= 1  OUTPUT: 1 • INPUT: x1 = 1, x2 = -1 • .75*1 + .75*-1 = 0 < 1  OUTPUT: -1 • INPUT: x1 = -1, x2 = 1 • .75*-1 + .75*1 = 0 < 1  OUTPUT: -1 • INPUT: x1 = -1, x2 = -1 • .75*-1 + .75*-1 = -1.5 < 1  OUTPUT: -1 this neuron computes the AND function

  7. In-class exercise • specify weights and thresholds to compute OR • INPUT: x1 = 1, x2 = 1 • w1*1 + w2*1 >=  OUTPUT: 1 • INPUT: x1 = 1, x2 = -1 • w1*1 + w2*-1 >=  OUTPUT: 1 • INPUT: x1 = -1, x2 = 1 • w1*-1 + w2*1 >=  OUTPUT: 1 • INPUT: x1 = -1, x2 = -1 • w1*-1 + w2*-1 <  OUTPUT: -1

  8. Normalizing thresholds • to make life more uniform, can normalize the threshold to 0 • simply add an additional input x0 = 1, w0 = - • advantage: threshold = 0 for all neurons wixi >=   -*1 +wixi >= 0

  9. Perceptrons • Rosenblatt (1958) devised a learning algorithm for artificial neurons given a training set (example inputs & corresponding desired outputs) • start with some initial weights • iterate through the training set, collect incorrect examples • if all examples correct, then DONE • otherwise, update the weights for each incorrect example if x1, …,xn should have fired but didn't, wi += xi (0 <= i <= n) if x1, …,xn shouldn't have fired but did, wi -= xi (0 <= i <= n) • GO TO 2 • artificial neurons that utilize this learning algorithm are known as perceptrons

  10. Example: perceptron learning • Suppose we want to train a perceptron to compute AND training set: x1 = 1, x2 = 1  1 x1 = 1, x2 = -1  -1 x1 = -1, x2 = 1  -1 x1 = -1, x2 = -1  -1 randomly, let: w0 = -0.9, w1 = 0.6, w2 = 0.2 using these weights: x1 = 1, x2 = 1: -0.9*1 + 0.6*1 + 0.2*1 = -0.1  -1 WRONG x1 = 1, x2 = -1: -0.9*1 + 0.6*1 + 0.2*-1 = -0.5  -1 OK x1 = -1, x2 = 1: -0.9*1 + 0.6*-1 + 0.2*1 = -1.3  -1 OK x1 = -1, x2 = -1: -0.9*1 + 0.6*-1 + 0.2*-1 = -1.7  -1 OK new weights: w0 = -0.9 + 1 = 0.1 w1 = 0.6 + 1 = 1.6 w2 = 0.2 + 1 = 1.2

  11. Example: perceptron learning (cont.) using these updated weights: x1 = 1, x2 = 1: 0.1*1 + 1.6*1 + 1.2*1 = 2.9  1 OK x1 = 1, x2 = -1: 0.1*1 + 1.6*1 + 1.2*-1 = 0.5  1 WRONG x1 = -1, x2 = 1: 0.1*1 + 1.6*-1 + 1.2*1 = -0.3  -1 OK x1 = -1, x2 = -1: 0.1*1 + 1.6*-1 + 1.2*-1 = -2.7  -1 OK new weights: w0 = 0.1 – 1 = -0.9 w1 = 1.6 – 1 = 0.6 w2 = 1.2 + 1 = 2.2 using these updated weights: x1 = 1, x2 = 1: -0.9*1 + 0.6*1 + 2.2*1 = 1.9  1 OK x1 = 1, x2 = -1: -0.9*1 + 0.6*1 + 2.2*-1 = -2.5  -1 OK x1 = -1, x2 = 1: -0.9*1 + 0.6*-1 + 2.2*1 = 0.7  1 WRONG x1 = -1, x2 = -1: -0.9*1 + 0.6*-1 + 2.2*-1 = -3.7  -1 OK new weights: w0 = -0.9 – 1 = -1.9 w1 = 0.6 + 1 = 1.6 w2 = 2.2 – 1 = 1.2

  12. Example: perceptron learning (cont.) using these updated weights: x1 = 1, x2 = 1: -1.9*1 + 1.6*1 + 1.2*1 = 0.9  1 OK x1 = 1, x2 = -1: -1.9*1 + 1.6*1 + 1.2*-1 = -1.5  -1 OK x1 = -1, x2 = 1: -1.9*1 + 1.6*-1 + 1.2*1 = -2.3  -1 OK x1 = -1, x2 = -1: -1.9*1 + 1.6*-1 + 1.2*-1 = -4.7  -1 OK DONE! EXERCISE: train a perceptron to compute OR

  13. Convergence • key reason for interest in perceptrons: Perceptron Convergence Theorem • The perceptron learning algorithm will always find weights to classify the inputs if such a set of weights exists. Minsky & Papert showed such weights exist if and only if the problem is linearly separable intuition: consider the case with 2 inputs, x1 and x2 if you can draw a line and separate the accepting & non-accepting examples, then linearly separable the intuition generalizes: for n inputs, must be able to separate with an (n-1)-dimensional plane.

  14. Linearly separable • why does this make sense? firing depends on w0 + w1x1 + w2x2 >= 0 border case is when w0 + w1x1 + w2x2 = 0 i.e., x2 = (-w1/w2) x1 + (-w0 /w2) the equation of a line the training algorithm simply shifts the line around (by changing the weight) until the classes are separated

  15. Inadequacy of perceptrons • inadequacy of perceptrons is due to the fact that many simple problems are not linearly separable however, can compute XOR by introducing a new, hidden unit

  16. Hidden units • the addition of hidden units allows the network to develop complex feature detectors (i.e., internal representations) • e.g., Optical Character Recognition (OCR) • perhaps one hidden unit • "looks for" a horizontal bar • another hidden unit • "looks for" a diagonal • the combination of specific • hidden units indicates a 7

  17. Building multi-layer nets • smaller example: can combine perceptrons to perform more complex computations (or classifications) • 3-layer neural net • 2 input nodes • 1 hidden node • 2 output nodes • RESULT? HINT: left output node is AND right output node is XOR FULL ADDER

  18. Hidden units & learning • every classification problem has a perceptron solution if enough hidden layers are used • i.e., multi-layer networks can compute anything • (recall: can simulate AND, OR, NOT gates) • expressiveness is not the problem – learning is! • it is not known how to systematically find solutions • the Perceptron Learning Algorithm can't adjust weights between levels • Minsky & Papert's results about the "inadequacy" of perceptrons pretty much killed neural net research in the 1970's • rebirth in the 1980's due to several developments • faster, more parallel computers • new learning algorithms e.g., backpropagation • new architectures e.g., Hopfield nets

  19. Backpropagation nets • backpropagation nets are multi-layer networks • normalize inputs between 0 (inhibit) and 1 (excite) • utilize a continuous activation function • perceptrons utilize a stepwise activation function output = 1 if sum >= 0 0 if sum < 0 • backpropagation nets utilize a continuous activation function output = 1/(1 + e-sum)

  20. Backpropagation example (XOR) • x1 = 1, x2 = 1 • sum(H1) = -2.2 + 5.7 + 5.7 = 9.2, output(H1) = 0.99 • sum(H2) = -4.8 + 3.2 + 3.2 = 1.6, output(H2) = 0.83 • sum = -2.8 + (0.99*6.4) + (0.83*-7) = -2.28, output = 0.09 • x1 = 1, x2 = 0 • sum(H1) = -2.2 + 5.7 + 0 = 3.5, output(H1) = 0.97 • sum(H2) = -4.8 + 3.2 + 0 = -1.6, output(H2) = 0.17 • sum = -2.8 + (0.97*6.4) + (0.17*-7) = 2.22, output = 0.90 • x1 = 0, x2 = 1 • sum(H1) = -2.2 + 0 + 5.7 = 3.5, output(H1) = 0.97 • sum(H2) = -4.8 + 0 + 3.2 = -1.6, output(H2) = 0.17 • sum = -2.8 + (0.97*6.4) + (0.17*-7) = 2.22, output = 0.90 • x1 = 0, x2 = 0 • sum(H1) = -2.2 + 0 + 0 = -2.2, output(H1) = 0.10 • sum(H2) = -4.8 + 0 + 0 = -4.8, output(H2) = 0.01 • sum = -2.8 + (0.10*6.4) + (0.01*-7) = -2.23, output = 0.10

  21. Backpropagation learning • there exists a systematic method for adjusting weights, but no global convergence theorem (as was the case for perceptrons) • backpropagation (backward propagation of error) – vaguely stated • select arbitrary weights • pick the first test case • make a forward pass, from inputs to output • compute an error estimate and make a backward pass, adjusting weights to reduce the error • repeat for the next test case testing & propagating for all training cases is known as an epoch • despite the lack of a convergence theorem, backpropagation works well in practice • however, many epochs may be required for convergence

  22. Problems/challenges in neural nets research • learning problem • can the network be trained to solve a given problem? • if not linearly separable, no guarantee (but backprop effective in practice) • architecture problem • are there useful architectures for solving a given problem? • most applications use a 3-layer (input, hidden, output), fully-connected net • scaling problem • how can training time be minimized? • difficult/complex problems may require thousands of epochs • generalization problem • how know if the trained network will behave "reasonably" on new inputs? • cross-validation often used in practice • split training set into training & validation data • after each epoch, test the net on the validation data • continue until performance on the validation data diminishes (e.g., hillclimb)

  23. Neural net applications • pattern classification • 9 of top 10 US credit card companies use Falcon • uses neural nets to model customer behavior, identify fraud • claims improvement in fraud detection of 30-70% • Sharp, Mitsubishi, … -- Optical Character Recognition (OCR) • prediction & financial analysis • Merrill Lynch, Citibank, … -- financial forecasting, investing • Spiegel – marketing analysis, targeted catalog sales • control & optimization • Texaco – process control of an oil refinery • Intel – computer chip manufacturing quality control • AT&T – echo & noise control in phone lines (filters and compensates) • Ford engines utilize neural net chip to diagnose misfirings, reduce emissions • recall from AI video: ALVINN project at CMU trained a neural net to drive • backpropagation network: video input, 9 hidden units, 45 outputs

  24. Interesting variation: Hopfield nets • in addition to uses as acceptor/classifier, neural nets can be used as associative memory – Hopfield (1982) • can store multiple patterns in the network, retrieve • interesting features • distributed representation • info is stored as a pattern of activations/weights • multiple info is imprinted on the same network • content-addressable memory • store patterns in a network by adjusting weights • to retrieve a pattern, specify a portion (will find a near match) • distributed, asynchronous control • individual processing elements behave independently • fault tolerance • a few processors can fail, and the network will still work

  25. Hopfield net examples • processing units are in one of two states: active or inactive • units are connected with weighted, symmetric connections positive weight  excitatory relation negative weight  inhibitory relation • to imprint a pattern • adjust the weights appropriately (algorithm ignored here) • to retrieve a pattern: • specify a partial pattern in the net • perform parallel relaxation to achieve a steady state representing a near match

  26. Parallel relaxation • parallel relaxation algorithm: • pick a random unit • sum the weights on connections to active neighbors • if the sum is positive  make the unit active if the sum is negative  make the unit inactive • repeat until a stable state is achieved • note: parallel relaxation = search • this Hopfield net has 4 stable states • parallel relaxation will start with an initial state and converge to one of these stable states

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